What is the Jacobian of the following function

Question

Consider a function F: $R^n \to R^n$ defined by $$f(u) = A*u*(n+1)+\lambda *B$$

Where A is a tridiagonal n-by-n matrix with -2 on the main diagonal and 1 on the off diagonals.
B = $\begin{pmatrix} { e }^{ { u }_{ 1 } } \\ ... \\ { e }^{ { u }_{ n } } \end{pmatrix}$

What is the Jacobian of F?

Answer 1

$\nabla f = A(n+1)+\lambda \mathrm{diag}(e^{u_1},e^{u_2},...,e^{u_n})$. ($\mathrm{diag}$ is building a diagonal matrix, with the diagonal elements given in the brackets)